Introduction to Robotics

Tutorial 7

Technion, cs department, Introduction to Robotics 236927

Winter 2010-2011

1

Potential Functions

2

1. Write the attraction and repulsion potential functions.

Destination

ObstacleCenter at (L,0)Radius = R

x

y

Destination

3

• The destination is modeled as an attractive charge.

Destination

x

y 22, yxyxU

ddU

A

A

-10-5

05

10

-10

-5

0

5

100

5

10

15

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Obstacle

4

• The Obstacle is modeled as a single repulsive charge.

22/,

/

yLxyxU

ddU

R

R

ObstacleCenter at (L,0)Radius = R

x

y

-10

-5

0

5

10

-10

-5

0

5

100

5

10

15

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Obstacle and Destination

5

yxUyxUyxU RA ,,,

-10-5

05

10

-10

-5

0

5

100

5

10

15

20

25

30

Obstacle and Destination

6

yxFyxFyxF RA ,,,

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

Potential Functions

7

2. For which α and β the robot will never hit the obstacle?

yyxLx

yLxyyxx

yx

yxFyxFyxF RA

ˆˆˆˆ

,,,

2/32222

Destination

ObstacleCenter at (L,0)Radius = R

x

y

Potential Functions

8

3. Will the robot always arrive at the destination?

4. From which starting positions the robot will not arrive the destination?

Different Obstacle Modeling

9

• The Obstacle is modeled as a single repulsive charge.

22

0

/

yLxd

else

RdRddU R

Potential Functions

10

5. For which α and β the robot will never hit the obstacle?

6. Will the robot always arrive at the destination?

7. From which starting positions the robot will not arrive the destination?

8. How does changing β effects the resulting path?

Different Obstacle Modeling

11

• The Obstacle is modeled as a single repulsive charge:

• Alternately:

Where d* is the distance to the closest point of the obstacle.

22

0

/

yLxd

else

RdRddU R

else

dddU R

0

0/ ***

Different Obstacle Modeling

12

else

dddU R

0

0/ ***

Another Example

13

Destination

x

y